Orthogonal determinants of GLn(q)

Abstract

Let n be a positive integer and q be a power of an odd prime. We provide explicit formulas for calculating the orthogonal determinants $\det \left(\chi \right)$, where $\chi \in \mathrm{Irr}\left({\mathrm{GL}}_n\right(q\left)\right)$ is an orthogonal character of even degree. Moreover, we show that $\det \left(\chi \right)$ is “odd”. This confirms a special case of a conjecture by Richard Parker.